the dimensions of a rectangle are such that its length is 5 in more than its width. If the length were doubled and the width were decreases by 2 in, the area would be increased 136 in^2,what are the length and width of the rectangle
Initially
w=width
w+5=length
w(w+5)=original area
Subsequently:
w-2=width
2(w+5)=length
(w-2)*2(w+5)=2*original area=w(w+5)=area
Therefore
2(w-2)(w+5)=w(w+5)
Solve for w:
2(w-2)=w (w cannot be equal to -5)
2w-4=w
w=4
original width -- x
original length -- x+5
area = x(x+5)
new width = x-2
new length = 2x+10
new area = (2x+10)(x-2)
(2x+10)(x-2) - x(x+5) = 136
2x^2 + 6x - 20 - x^2 - 5x - 136 = 0
x^2 + x - 156 = 0
(x-12)(x+13) = 0
x = 12 or x = -13
ignoring the negative,
width = 12
length = 17
check:
original area = 12(17) = 204
new width = 10
new length = 34
new area = 10(34) = 340
increase in area = 340-204 = 136 , YEahh!
Good catch!
Thank you!
To solve this problem, we can use algebraic equations. Let's assign variables to represent the width and length of the rectangle.
Let's call the width of the rectangle 'w' (in inches).
According to the problem, the length is 5 inches more than the width, so we can represent the length as 'w + 5' (in inches).
Now, let's set up the equation for the original area of the rectangle using the given dimensions:
Area = Length * Width
Area = (w + 5) * w
Next, we are told that if the length is doubled (2 * (w + 5)) and the width is decreased by 2 inches (w - 2), the new area would be increased by 136 square inches. Thus, the equation for the new area would be:
New Area = (2 * (w + 5)) * (w - 2)
Now we can set up an equation using the areas:
New Area - Original Area = 136
((2 * (w + 5)) * (w - 2)) - ((w + 5) * w) = 136
Simplifying the equation, we get:
((2w + 10)(w - 2)) - (w^2 + 5w) = 136
Now we can solve this quadratic equation for 'w':
2w^2 - 4w + 10w - 20 - w^2 - 5w = 136
w^2 + w - 20 - 136 = 0
w^2 + w - 156 = 0
Now we can factor this quadratic equation:
(w + 13)(w - 12) = 0
Setting each factor equal to zero, we have:
w + 13 = 0 or w - 12 = 0
w = -13 or w = 12
Since the width cannot be negative, we disregard w = -13.
Therefore, the width of the rectangle is 12 inches.
To find the length, we can substitute the width value back into the equation for the length:
Length = Width + 5 = 12 + 5 = 17 inches
Hence, the length of the rectangle is 17 inches and the width is 12 inches.